Improving the Solution Procedure of Incremental Harmonic Balance Method for Multi-Degree-of-Freedom Self-Excited Vibration Systems

When applying the Incremental Harmonic Balance (IHB) method to solve the limit cycle in a self-excited system, the practice of fixing one of the harmonic coefficients effectively solves the problem that there are more unknowns than algebraic equations due to the presence of unknown angular frequencies in the periodic response. However, an inappropriate value for this fixed harmonic coefficient can significantly affect the convergence of subsequent Newton–Poisson iterations. To address this issue, we propose a technique that dynamically adjusts the value of the fixed harmonic coefficient, effectively improving the convergence and stability of the traditional IHB method and providing an efficient solution for the parametric study of self-excited systems. The efficiency and performance of our method in accurately predicting periodic solutions are demonstrated by its application to Van der Pol oscillators and predicting vortex-induced vibrations in cable-stayed structures. Numerical simulations show that the results obtained by the proposed method are very close to those obtained by the Runge–Kutta method. This study opens up a robust and efficient approach for analyzing and studying the dynamic behavior of complex self-excited systems.